Lecture Notes in Differential Equations - Bruce E. Shapiro

132 LESSON 15. LINEAR OPERATORS AND VECTOR SPACES
Definition 15.4. If y, z are both elements of a vector space V, and A and
B are any numbers, we call
a Linear Combination of y and z
w = Ay + Bz (15.41)
Example 15.8. If v = (1, 0, 3) and w = (5, −3, 12) are vectors in Euclidean
3 space, then for any numbers A and B,
u = Av + Bw = A(1, 0, 3) + B(5, −3, 12) = (A + 5B, −3B, 3A + 12B)
(15.42)
is a linear combination of v and w.
The closure property of vector spaces is sometimes stated as following: Any
linear combination of vectors is an element of the same vector
space. For example, we can create linear combinations of functions and
we know that they are also in the same vector space.
Example 15.9. Let f(t) = 3 sin t, g(t) = t 2 − 4t be functions in the vector
space V of real valued functions. Then if A and B are any real numbers,
h(t) = Af(t) + Bg(t) (15.43)
= A sin t + B(t 2 − 4t) (15.44)
is a linear combination of the functions f and g. Since V is a vector space,
h is also in V.
Example 15.10. Let V be the vector space consisting of real functions on
the real numbers. Then differentiation, defined by
D(y) = dy(t)
dt
(15.45)
is a linear operator. To see that both properties hold let y(t) and z(t) be
functions (e.g., y, z ∈ V) and let c be a constant. Then
D(y + z) =
d(y(t) + z(t)
dt
= dy(t)
dt
D(cy) = d(cy(t)
dt
Hence D is a linear operator.
+ dz(t)
dt
= c dy(t)
dt
= D(y) + D(z) (15.46)
= cD(y) (15.47)